We have shown that the angle sum of a triangle is . We have also shown that the measure of an inscribed angle is half the measure the central angle that intercepts the same arc. In this post, we use the Inscribed Angle Theorem to show that the Triangle Angle Sum Theorem holds.

Theorem

The angle sum of a triangle is .

Proof

Consider the figures above. In the first figure, the triangle is divided into three central angles. Clearly the three angles add up to a complete rotation about the center so

In the second figure, the colored angles are inscribed angles. The measure of each angle is half the measure of its corresponding central angle (the angle of the same color) in the first figure (see also third figure). That is, the measure of the blue angle in the second figure is half the measure of the blue angle in the first figure. This means that the sum of measures of the inscribed angles equal to